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μI
	
  Taking	
  the	
  perturbative	
  expansion	
  with	
  	
  	
  	
  	
  	
  for	
  the	
  contribution	
  of
quark	
  loop,	
  resulting	
  in	
  deriving	
  the	
  relation	
  of	
  GL	
  couplings
where	
  	
  	
  	
  	
  	
  	
  refers	
  to	
  the	
  quantity	
  in	
  the	
  absence	
  of	
  	
  	
  	
  	
  	
  	
  and	
  	
  is	
  taken	
  away
in	
  following	
  text
µI
Structure change of the chiral critical point driven by isospin density
Department of Physics, Tokyo University of Science, Tokyo 162-8601, Japan
QCD@Work @ Lecce - Italy, 18-21 June 2012
Yuhei Iwata, Hiroaki Abuki and Katsuhiko Suzuki
1. Introduction
Abstract
4. Summary
	
  	
  We	
  study	
  the	
  influence	
  of	
  the	
  isospin	
  asymmetry	
  in	
  strong	
  interacting	
  quark	
  matters	
  near	
  the	
  chiral	
  tricritical	
  point	
  (TCP).	
  At	
  the	
  vicinity	
  of	
  TCP	
  
we	
  find	
  the	
  drastic	
  change	
  of	
  the	
  phase	
  diagram:	
  shifting	
  of	
  TCP	
  and	
  splitting	
  of	
  TCP	
  into	
  new	
  four	
  critical	
  points.	
  Accordingly	
  the	
  homogeneous	
  
pion	
  condensate	
  and	
  solitonic	
  one	
  come	
  to	
  occupy	
  	
  large	
  domains	
  in	
  the	
  phase	
  diagram.
Reference
	
  	
  We	
  showed	
  how	
  the	
  isospin	
  asymmetry	
  affects	
  the	
  phase	
  structure	
  
near	
  the	
  TCP	
  and	
  found	
  the	
  remarkable	
  effects,	
  e.g.	
  a	
  shift	
  of	
  TCP	
  
location	
  and	
  splitting	
  of	
  TCP.	
  Further,	
  we	
  found	
  the	
  homogeneous	
  
and	
  solitonic	
  pion	
  condensates	
  have	
  large	
  region	
  instead	
  of	
  chiral	
  
condensates	
  on	
  the	
  phase	
  diagram	
  of	
  GL	
  coupling	
  space.
[1]	
  	
  R.	
  Casalbuoni,	
  PoS	
  CPOD	
  2006,	
  001	
  (2006).
[2]	
  D.	
  Nickel,	
  Phys.	
  Rev.	
  Lett.	
  103,	
  072301	
  (2009).
[3]	
  H.	
  Abuki,	
  D.	
  Ishibashi	
  and	
  K.	
  Suzuki,	
  Phys.	
  Rev.	
  D	
  85,	
  074002	
  
(2012).
[4]	
  A.	
  I.	
  Buzdin	
  and	
  H.	
  Kachkachi,	
  Phys.	
  Lett.	
  A225,	
  341	
  (1997).
2. Method
3. Results
our objective
	
  We	
  investigate	
  how	
  the	
  phase	
  structures	
  near	
  TCP	
  changes	
  at	
  finite	
  	
  	
  	
  	
  
focusing	
  on	
  the	
  isospin	
  density
µI
In	
  the	
  result	
  of	
  variation	
  :	
  
	
  Using	
  GL	
  analysis	
  for	
  describing	
  phase	
  structures	
  around	
  TCP
solid	
  line	
  :	
  1st	
  order
dashed	
  line:	
  	
  2nd	
  order
mapping
Phase	
  diagram	
  with	
  
inhomogeneous	
  states	
  included
.Phase	
  diagram	
  with	
  
homogeneous	
  states	
  only
.
	
  fine	
  structures	
  appear	
  
below	
  TCP’
	
  solitonic	
  pion	
  condensate	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  
dominate	
  the	
  fine	
  structures
	
  at	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  ,	
  a	
  major	
  part	
  of
solitonic	
  σ(x)	
  taken	
  over	
  by	
  π(x)
CEP
χSB
(σ)
Wigner
T
μq
1st	
  order
crossover
our	
  focus
μI
?µI = µu µd
mq = 0
QCD	
  phase	
  diagram
1st	
  order
crossover
2nd	
  order
in	
  GL	
  parameter	
  space
	
  We	
  study	
  the	
  effect	
  of	
  a	
  finite	
  quark	
  mass	
  and	
  an	
  isospin	
  asymmetry	
  
for	
  realistic	
  systems	
  on	
  the	
  chiral	
  tricritical	
  point	
  (TCP)[1]
	
  The	
  quark	
  mass	
  turns	
  TCP	
  into	
  CEP	
  simply,	
  but	
  doesn’t	
  change	
  phases
	
  The	
  isospin	
  asymmetry	
  which,	
  for	
  example,	
  arises	
  from	
  the	
  neutrality	
  
affects	
  phase	
  structures	
  at	
  vacuum:	
  the	
  emergence	
  of	
  pion	
  condensate
focus	
  on	
  the	
  isospin	
  density
= ( , 1, 2, 3)order	
  parameter	
  :	
  chiral	
  four	
  vector	
  
c
	
  Assume	
  :	
  quark	
  loop	
  contribution	
  is	
  dominant	
  over	
  gluonic	
  one
	
  	
  Justified:	
  if	
  TCP	
  is	
  located	
  at	
  large	
  fugacity	
  region eµc/Tc
1
Ginzburg-Landau potential.
Perturbative expansion.
	
  π	
  occupies	
  large	
  domain	
  
in	
  the	
  phase	
  diagram
	
  bicritical	
  point	
  P	
  appears	
  
among	
  π,	
  σ,	
  Wigner	
  phases.
	
  TCP	
  shifts	
  to	
  (0, µ2
I)
GL[ , 0] =
2
2
2
+
4 + µ2
I
4
4
+
1
6
6
.
0 0
	
  TCP’	
  changes	
  into	
  a	
  Lifshitz
point	
  and	
  splits	
  into	
  three
critical	
  points	
  :	
  P,	
  Q,	
  R
4 µ2
I
	
  	
  	
  	
  	
  -­‐term	
  favors	
  the	
  pion
condensate	
  π	
  with
(x) = k sn(kz; ) ,
µI
2
4 > 0
(0)
2
4
6
=
1 aµ2
I O(µ4
I)
0 1 bµ2
I
0 0 1
(0)
2
(0)
4
(0)
6
,
2
4 [or 4b(c)]
=
cµ2
I O(µ4
I)
0 dµ2
I [db(c)µ2
I]
(0)
4
(0)
6
.
SU(2)L SU(2)R
SU(2)L SU(2)R
U(1)I3;V U(1)I3;A
Chiral	
  symmetry
Explicit	
  breaking
of	
  symmetry
h
ignoring	
  quark	
  mass	
  term,
considering	
  in	
  future
GL[ (x), (x)] =
2
2
2
+
4
4
( 2
)2
+
4,b
4
( )2
+
6
6
( 2
)3
+
6,b
6
( , )2
+
6,c
6
[ 2
( )2
( , )2
] +
6,d
6
( )2
2
2
2
c
4
4
4
c
4,b
4
( 2 2
c ) 2
c
4,c
4
( 2
c )2
a = 0, b = 1, c = 1/2, d = db(c) = 1
	
  For	
  chiral	
  symmetric	
  term[2] 4,b = 4 , ( 6,b, 6,c, 6,d) = (5, 3, 1/2) 6
arXiv:1206.2870
U(1)I3;V U(1)I3;A
U(1)I3;V
U(1)I3;A
solitonic solitonic
U(1)I3;V U(1)I3;A
U(1)I3;V
U(1)I3;A
[4]
	
  Let	
  us	
  discuss	
  the	
  phase	
  diagram	
  in	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  -­‐	
  space
	
  We	
  get	
  the	
  GL	
  potential	
  
parametrized	
  by	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  
and	
  scale	
  every	
  quantity	
  in	
  the	
  unit	
  	
  	
  	
  	
  	
  	
  	
  ,	
  having	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  [3]6 = 1[ 6]
{ 2, 4, 6, µ2
I}
( 2, 4, µ2
I)

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Structure change of the chiral critical point driven by isospin density (QCD@Work2012, poster)

  • 1. μI  Taking  the  perturbative  expansion  with            for  the  contribution  of quark  loop,  resulting  in  deriving  the  relation  of  GL  couplings where              refers  to  the  quantity  in  the  absence  of              and    is  taken  away in  following  text µI Structure change of the chiral critical point driven by isospin density Department of Physics, Tokyo University of Science, Tokyo 162-8601, Japan QCD@Work @ Lecce - Italy, 18-21 June 2012 Yuhei Iwata, Hiroaki Abuki and Katsuhiko Suzuki 1. Introduction Abstract 4. Summary    We  study  the  influence  of  the  isospin  asymmetry  in  strong  interacting  quark  matters  near  the  chiral  tricritical  point  (TCP).  At  the  vicinity  of  TCP   we  find  the  drastic  change  of  the  phase  diagram:  shifting  of  TCP  and  splitting  of  TCP  into  new  four  critical  points.  Accordingly  the  homogeneous   pion  condensate  and  solitonic  one  come  to  occupy    large  domains  in  the  phase  diagram. Reference    We  showed  how  the  isospin  asymmetry  affects  the  phase  structure   near  the  TCP  and  found  the  remarkable  effects,  e.g.  a  shift  of  TCP   location  and  splitting  of  TCP.  Further,  we  found  the  homogeneous   and  solitonic  pion  condensates  have  large  region  instead  of  chiral   condensates  on  the  phase  diagram  of  GL  coupling  space. [1]    R.  Casalbuoni,  PoS  CPOD  2006,  001  (2006). [2]  D.  Nickel,  Phys.  Rev.  Lett.  103,  072301  (2009). [3]  H.  Abuki,  D.  Ishibashi  and  K.  Suzuki,  Phys.  Rev.  D  85,  074002   (2012). [4]  A.  I.  Buzdin  and  H.  Kachkachi,  Phys.  Lett.  A225,  341  (1997). 2. Method 3. Results our objective  We  investigate  how  the  phase  structures  near  TCP  changes  at  finite           focusing  on  the  isospin  density µI In  the  result  of  variation  :    Using  GL  analysis  for  describing  phase  structures  around  TCP solid  line  :  1st  order dashed  line:    2nd  order mapping Phase  diagram  with   inhomogeneous  states  included .Phase  diagram  with   homogeneous  states  only .  fine  structures  appear   below  TCP’  solitonic  pion  condensate                     dominate  the  fine  structures  at                                        ,  a  major  part  of solitonic  σ(x)  taken  over  by  π(x) CEP χSB (σ) Wigner T μq 1st  order crossover our  focus μI ?µI = µu µd mq = 0 QCD  phase  diagram 1st  order crossover 2nd  order in  GL  parameter  space  We  study  the  effect  of  a  finite  quark  mass  and  an  isospin  asymmetry   for  realistic  systems  on  the  chiral  tricritical  point  (TCP)[1]  The  quark  mass  turns  TCP  into  CEP  simply,  but  doesn’t  change  phases  The  isospin  asymmetry  which,  for  example,  arises  from  the  neutrality   affects  phase  structures  at  vacuum:  the  emergence  of  pion  condensate focus  on  the  isospin  density = ( , 1, 2, 3)order  parameter  :  chiral  four  vector   c  Assume  :  quark  loop  contribution  is  dominant  over  gluonic  one    Justified:  if  TCP  is  located  at  large  fugacity  region eµc/Tc 1 Ginzburg-Landau potential. Perturbative expansion.  π  occupies  large  domain   in  the  phase  diagram  bicritical  point  P  appears   among  π,  σ,  Wigner  phases.  TCP  shifts  to  (0, µ2 I) GL[ , 0] = 2 2 2 + 4 + µ2 I 4 4 + 1 6 6 . 0 0  TCP’  changes  into  a  Lifshitz point  and  splits  into  three critical  points  :  P,  Q,  R 4 µ2 I          -­‐term  favors  the  pion condensate  π  with (x) = k sn(kz; ) , µI 2 4 > 0 (0) 2 4 6 = 1 aµ2 I O(µ4 I) 0 1 bµ2 I 0 0 1 (0) 2 (0) 4 (0) 6 , 2 4 [or 4b(c)] = cµ2 I O(µ4 I) 0 dµ2 I [db(c)µ2 I] (0) 4 (0) 6 . SU(2)L SU(2)R SU(2)L SU(2)R U(1)I3;V U(1)I3;A Chiral  symmetry Explicit  breaking of  symmetry h ignoring  quark  mass  term, considering  in  future GL[ (x), (x)] = 2 2 2 + 4 4 ( 2 )2 + 4,b 4 ( )2 + 6 6 ( 2 )3 + 6,b 6 ( , )2 + 6,c 6 [ 2 ( )2 ( , )2 ] + 6,d 6 ( )2 2 2 2 c 4 4 4 c 4,b 4 ( 2 2 c ) 2 c 4,c 4 ( 2 c )2 a = 0, b = 1, c = 1/2, d = db(c) = 1  For  chiral  symmetric  term[2] 4,b = 4 , ( 6,b, 6,c, 6,d) = (5, 3, 1/2) 6 arXiv:1206.2870 U(1)I3;V U(1)I3;A U(1)I3;V U(1)I3;A solitonic solitonic U(1)I3;V U(1)I3;A U(1)I3;V U(1)I3;A [4]  Let  us  discuss  the  phase  diagram  in                                          -­‐  space  We  get  the  GL  potential   parametrized  by                                                         and  scale  every  quantity  in  the  unit                ,  having                              [3]6 = 1[ 6] { 2, 4, 6, µ2 I} ( 2, 4, µ2 I)